I used to pick up my wife at her job and pretty well won a prize every time I played, if the prize was in the right position and worth getting.

Depends on the machine - some let you drop the claw a little bit at a time, which is what the one I played did. I would drop it down so that eventually the claw enveloped the targeted toy. Then I would drop it a bit more so the claw was still wrapped around the toy, but resting on its side. When time ran out or I held down the button, the claw would start to close, but it would be able to wrap around the toy before the claw got lifted.

It also helps to try and pick a loose toy so the claw can wrap it properly, as well as end up on its side.